{
  "id": 2667,
  "curve_key": "129:351",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-3",
    "0"
  ],
  "rank_lower_bound": 2,
  "torsion": [],
  "naive_height": 14.579437213085017,
  "faltings_height": -0.7203885829002808,
  "conductor": "1171",
  "bad_primes": [
    "1171"
  ],
  "discriminant": "1171",
  "regulator": "0.336702983678594590092633836904224732983202",
  "points": [
    [
      "0",
      "0"
    ],
    [
      "-1",
      "1"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 1171a1 = LMFDB 1171.a1: https://www.lmfdb.org/EllipticCurve/Q/1171/a/1\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 1171, minimal discriminant 1171, torsion trivial. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:16:46",
  "updated_at": "2026-09-26 20:16:46"
}